Friction reduced toward infinity; a giant calculation
Stacked wheels turned by a falling weight, with a geometric progression of friction
This is the climax of the anti-friction study: a tall column of wheels of increasing size, each on an easy pivot, driven by a cord wound 30 or 40 times around the top wheel n with a weight m hung from its end. Leonardo claims that once released the weight will spin the wheel so freely that it turns by itself for nearly a whole long day. Beside it he runs the numbers, a geometric progression 1, 24, 576, 13,824 ... up to 110,075,314,176, to show that one pound of friction at the weight m stands balanced against more than a hundred thousand million pounds of friction saved. His governing rule, written at the foot, is that every friction resists with a quarter of the weight being rubbed. A ratchet-and-pawl saw device is drawn at the lower left, and he notes this eighth proposition belongs in his book on impetus.
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Friction diminished toward infinity by added wheels
Titled a 'process extensible toward infinity of the diminution of the power of friction,' the design stacks wheels each larger than the one below so the edge of a lower wheel never impedes the pivot of the one above. A weight m on a cord wound many times around the top wheel n, once let fall, spins that wheel so easily it turns by itself for nearly a very great day; the wheels are to be thin, truly round, half a braccio wide, with polished, strong, thin pivots.
One pound of friction against 110,075,314,176
With each wheel a braccio across and the pivot half an ounce thick, giving a quarter-ounce of counter-leverage against the cord's friction, Leonardo compounds the advantage down the column as a geometric progression, 1, 24, 576, 13,824, up to 110,075,314,176. One pound of friction at the weight m thus stands equal to more than a hundred thousand million pounds of friction.
Every friction resists a quarter of the weight rubbed
The rule underpinning the whole calculation, set at the lower left: every friction resists by a fourth of the weight that is rubbed. This constant coefficient lets each stage of the wheel column be multiplied to reach the enormous total.
Cord, weight and free-spinning wheel n
The apparatus is drawn plainly: wheel n at the top carries a cord wound thirty or forty times with the loose weight m hanging below, and the descending progression of wheels sits on easy pivots so the shaft runs almost unresisted. A toothed ratchet mechanism appears at the lower left.